The Fibration of Physical Law: Why String Theory and LQG Fail the Categorical Test

J. Rogers, SE Ohio

Abstract:
Modern quantum gravity research focuses on deriving the numerical values of dimensionful constants (

        $G, \hbar, l_s$
      

) from fundamental principles. We demonstrate via the Grothendieck Fibration framework that this objective is a category error. Physical laws are Cartesian liftings of dimensionless morphisms in a base category

        $\mathcal{B}$
      

to a total category of measurement

        $\mathcal{E}$
      

. The so-called "fundamental constants" are merely cocycle data (connection coefficients) required to maintain diagram commutativity between coordinate charts. By analyzing String Theory and Loop Quantum Gravity through the lens of Projection Calculus, we prove that both theories fail because they mistake the fiber for the base: String Theory hardcodes a coordinate artifact (

        $l_s$
      

) as an axiom, while LQG attempts to quantize the Jacobian of the measurement transformation itself (

        $1/G$
      

).

---

1. Introduction: The Categorical Error

Theoretical physics assumes that the "Theory of Everything" resides within the category of measured quantities,

        $\mathcal{E}$
      

. It seeks a Lagrangian that, when solved, outputs the specific values of the Planck scale or the gravitational constant.

This paper argues that

        $\mathcal{E}$
      

is the wrong locus of inquiry.

        $\mathcal{E}$
      

is merely the total space of a fibration

        $\pi: \mathcal{E} \to \mathcal{B}$
      

, where

        $\mathcal{B}$
      

is the category of dimensionless conceptual relationships. Real physics happens in

        $\mathcal{B}$
      

; the equations we write in

        $\mathcal{E}$
      

(like

        $E=m{c}^2$      

) are merely Cartesian liftings of base truths (

        $E \sim M$
      

) through a specific unit coordinate system.

The failure of String Theory and Loop Quantum Gravity is not technical; it is structural. They are attempting to derive the structure of the bundle by analyzing the artifacts of a specific local section.

2. The Grothendieck Fibration Framework

We define the structure of physical law as follows:

1. The Base Category (

           $\mathcal{B}$
         

   ): The realm of pure relational concepts (Mass, Time, Length). Morphisms here are unit-independent proportionalities (e.g.,

           $\phi: \text{Time} \to \text{Mass}^{-1}$
         

   ).

2. The Total Category (

           $\mathcal{E}$
         

   ): The realm of concrete measurements (numbers with units).

3. The Fibration (

           $\pi$
         

   ): The functor projecting measurements to their conceptual types.

4. The Constants as Cocycles: Constants like

           $c$
         

   and

           $G$
         

   are not objects in

           $\mathcal{B}$
         

   . They are transition functions in the fiber.

   •         $c$
           

     is the Jacobian rotating the Time axis to the Length axis.

   •         $\hbar$
           

     is the Jacobian rotating Frequency to Energy.

The "Calculus of Physical Law" (Rogers, 2025) demonstrates that any physical formula in

        $\mathcal{E}$
      

can be automatically compiled from a dimensionless relation in

        $\mathcal{B}$
      

combined with these Jacobian rotations.

3. Critique of String Theory: The Fixed-Scale Fallacy

String Theory posits a fundamental 1-dimensional object with a characteristic length scale

        $l_s$
      

. From the perspective of the Fibration Framework, this is a fatal methodology.

3.1 Hardcoding the Fiber

By assuming

        $l_s$
      

exists a priori, String Theory selects a specific coordinate chart in

        $\mathcal{E}$
      

and elevates it to ontology.

        $l_s$
      

is effectively a "Planck-scale meter stick." In a truly fundamental theory—one that describes the substrate

        $\mathcal{S}_u$
      

—length must be an emergent property of dimensionless relations, not an input parameter.

3.2 The Landscape as Coordinate Confusion

The "Landscape Problem" (

        $10^{500}$
      

vacua) is interpreted here not as a multiverse of physical realities, but as a cohomological redundancy. The moduli spaces of Calabi-Yau compactifications represent different ways to parametrize the fiber

        $\pi^{-1}(X)$
      

. Because String Theory lacks a coordinate-free description of the base

        $\mathcal{B}$
      

, it cannot distinguish between a change in physics and a change in the measurement gauge. The theory is drowning in unit equivalences it mistakes for distinct universes.

Verdict: String Theory is a "Newtonian" theory of the Planck scale—it assumes a background metric (the string length) rather than deriving the concept of distance from the dimensionless substrate.

4. Critique of Loop Quantum Gravity: Quantizing the Jacobian

Loop Quantum Gravity (LQG) attempts to quantize the geometry of spacetime directly, starting from the Einstein-Hilbert action which contains

        $1/16\pi G$
      

.

4.1 The 1/G Error

In the Projection Calculus,

        $G$
      

is identified as a composite Jacobian:

        $$G \sim \frac{\partial(\text{Length}^3 \cdot \text{Time}^{-2})}{\partial(\text{Mass})}$$      

        $G$
      

is the conversion factor that allows us to speak of Mass and Spacetime curvature in the same equation. It is a scaling artifact of using the SI (or any non-Planck) chart.

When LQG quantizes the area operator and derives spectra proportional to

        $l_P^2$
      

, it is essentially quantizing the conversion factor. It finds discreteness not necessarily in the substrate, but in the "pixels" of the coordinate map defined by

        $G$
      

,

        $\hbar$
      

, and

        $c$
      

.

4.2 Circular Prediction

LQG celebrates the derivation of the Area Spectrum:

        $A_j = 8\pi \gamma l_P^2 \sqrt{j(j+1)}$

.
However,

        $l_P$
      

is defined as

        $\sqrt{\hbar G/c^3}$

.
The theory inputs

        $\hbar, G, c$
      

(the Jacobians) and outputs a length scale dependent on them. This is tautological. It predicts that "if you measure geometry using Planck rulers, you get Planck answers." It fails to explain why the relationship between Mass and Geometry requires the specific rotation

        $G$
      

in the first place.

Verdict: LQG confuses the map for the territory. It models the granularity of the measurement apparatus (the unit system) rather than the continuous topology of the base category

        $\mathcal{B}$
      

.

5. The Evidence: The Law Compiler

Using the Law Compiler functor

        $\Lambda$
      

, we can empirically demonstrate the redundancy of these theories.

Consider the Hawking Temperature formula.

1. Base Truth (

           $\mathcal{B}$
         

   ):

           $T \sim 1/M$
         

   (Temperature is inversely proportional to Mass).

2. Projection: The Compiler

           $\Lambda$
         

   lifts this morphism to the SI fiber.

3. Jacobian Application: To equate a Temperature unit (

           $K$
         

   ) to an inverse Mass unit (

           $kg^{-1}$
         

   ), the system must apply the rotation sequence:
   

           $$K \xrightarrow{k_B^{-1}} E \xrightarrow{c^{-2}} M \xrightarrow{\text{inv}} M^{-1} \dots$$

4. Output:

           $T = \frac{\hbar c^3}{8\pi G k_B M}$
         

The constants appear automatically as the cost of doing business in SI units. String Theory and LQG spend decades trying to derive this formula, but the formula is just the "shadow" of the simple relation

        $T \sim 1/M$
      

cast through the lens of human history.

The conclusion is stark: Any theory that treats the constants as physical entities to be predicted, rather than algebraic rotation coefficients to be applied, is solving a malformed problem.

6. Conclusion: The Way Forward

The era of "Deriving the Constants" must end. The "Coordinate Artifact Problem" reveals that

        $G, c, \hbar$
      

are human choices encoded in nature's dimensional limits.

A successful theory of Quantum Gravity will not be found in

        $\mathcal{E}$
      

. It will be a theory of topological constraints on

        $\mathcal{B}$
      

. It must explain:

1. Why the Base Category allows the morphism

           $T \to 1/M$
         

   (Black Hole Thermodynamics).

2. Why the dimensionless ratio

           $m_p/m_e \approx 1836$
         

   is stable.

3. Why the fine-structure constant

           $\alpha$
         

   (a pure scalar in

           $\mathcal{B}$
         

   ) has its specific value.

String Theory and LQG have failed because they are "Fiber Theories"—obsessed with the size of the ruler (

        $l_s, l_P$
      

) rather than the shape of the relations. We must elevate our perspective to the Grothendieck base, where the laws of physics exist in their naked, unit-free form.